$ A = \left[\begin{array}{r}4 \\ 2 \\ -2\end{array}\right]$ $ B = \left[\begin{array}{rrr}3 & 4 & 0 \\ 1 & 4 & 0\end{array}\right]$ Is $ A B$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ A$ , have? How many rows does the second matrix, $ B$ , have? Since $ A$ has a different number of columns (1) than $ B$ has rows (2), $ A B$ is not defined.